The Lincoln Park Zoo gets very crowded, especially on nice days in the summer. It has an important advantage compared to many other museums, however: with free admission, you don't need to wait in any lines to buy tickets or get in.
Parking
Big crowds will especially impact parking, creating long lines just to get into the lot, and forcing guests to park well past the main entrances. The bus drop off is an entrance anyone can use, so consider that if you end up to the south of the zoo (the East Gate is a better entrance overall, closer to great animal exhibits and guest services, so I'd recommend heading that direction if you're in between the East gate the bus drop off).
If you're expecting big crowds and can get to the zoo with public transit or ride-shares, going in from the west side (the West Gate, North Gate, or Brauer Gate) is much easier than the east side parking lots.
Food
As with any museum, big crowds means long lines for food. Solve that by eating early: I recommend getting your food by 11:30 even on fairly calm days, and you may want to aim closer to 11 on the busiest days. Consider bringing your own food and eating at the tables or lawn in front of the Park Place Cafe, or in the Foreman Pavilion (though the latter is sometimes closed to the public); even with your own food, starting lunch by 11:30 on busy days will make it easier to find a spot to sit.
Exhibits
The Lincoln Park Zoo isn't huge, and can feel packed when it's crowded. The
play areas in the barn don't often get over-full, but they are really not enjoyable when they are. The
Center for African Apes will get very full but it's still worth squeezing into because it's so good. The
African Journey interiors might be overly crowded, but the zebras, giraffes, and rhinos along the outside are probably the best part and will handle the crowds better, along with the adjacent Arctic Tundra. For other exhibits, try some less popular alternatives, like the Bird House and the Small Mammal and Reptile house, both of which are good, if lacking in particularly exciting animals.